Chern-Simon level quantization and quantum Hall effect It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is depend on level quantization of Chern-Simon theory.
$$
\frac{k}{4\pi}\int_W AdA
$$
where $W$ is our three manifold. As I saw in literatures, for example at page 151, https://arxiv.org/abs/1606.06687, they look at the partition function of the theory at the finite temperature and take the spatial directions as a sphere $S^2$. So the total $2+1$ space-time is $W=S^2*S^1$ and it is easy to see that the level $k$ must be an integer for consistency of the quantum theory if we have a unit flux threading through the sphere.
$$
\frac{1}{2\pi}\int_{S^2} F=\frac{h}{e}
$$
my question is that, in real situation, our sample is not a sphere and usually is a rectangular and (I think) experiments are performed at zero temperature. How Chern-Simon level have to be quantized in $W=\mathbb{R^2}*\mathbb{R}$ background.
 A: The rectangle case is easier than the sphere as one can take $$\Phi_x= \int_{\alpha} A_x dx\\\Phi_y= \int_\beta A_y dy$$
as the gauge invariant degress of freedom. Here $\alpha$ and $\beta$ are the one-cycles generating $H_1[T^2]$. Then, if I remember correctly (it's been a while) the algebra of the commutator
$$
[\Phi_x,\Phi_y]= \frac{2\pi i}{k} 
$$ together with the compactness of the range of the $\Phi$'s requires an integer $k$.
The phase space is itself a torus and so the Hilbert space is finite dimensional and can be spanned by Theta functions of level $k$.
A recent reference is this.
A: (this is an argument coming from path integral quantization. There are more succinct arguments coming from canonical quantization).
The quantization of the Chern-Simons level comes from a requirement of gauge invariance at the quantum level. One can show that the variation of the action is something like $$\delta S= \text{boundary term} + 2\pi i k *\text{winding number of gauge transformation}$$
When $M=\mathbb{R}^3$, there are no large gauge transformations (that is, the winding number of all transformations is always zero) so $k$ can be any real number. This is unlike the case of other spacetimes, such as the one you are describing, where the winding number can be any integer, hence we require the quantization of $k$ to have a variation that is a integer multiple of $2\pi i$, so that the partition function does not change.
